e_311330
plain-language theorem explainer
Pointwise kernel identity: the folded coupling numerator at multi-index (3,1,1,3,3,0) equals eight times the explicit integer table entry. One of 256 decide-certified cells in chunk 13 of the 4D Regge midpoint M2TT identity. The assembly theorem cites it when discharging the universal quantification by fin_cases. Proof is a single kernel decide.
Claim. For the multi-index $(a,b,c,d,i,j)=(3,1,1,3,3,0)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(3,1,1,3,3,0)=8\,Z_{\mathrm{expl}}(3,1,1,3,3,0)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is the integer obtained by folding a fixed coupling list: each term contributes via a local kernel, and the fold starts at zero. The companion table $Z_{\mathrm{expl}}$ is an explicit six-index integer function on $\mathbb{F}_4$, with sparse nonzero values (e.g. $\pm 2,4$ on selected index patterns).
The module goal is the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on all $4^6=4096$ index tuples, discharged in 256-cell decide chunks. This declaration is cell 13 of that grid for the concrete indices $(3,1,1,3,3,0)$.
Upstream, both sides are pure definitions in the kernel certificate module: the fold for the numerator and the pattern-matched table for the closed form.
proof idea
One-line computational proof: decide evaluates both integer sides at the fixed Fin-4 indices and checks equality in the kernel. No lemmas are invoked; the definitions of the folded numerator and the explicit table reduce to concrete Int values that match.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{expl}}(a,b,c,d,i,j)$ and proves it by exhaustive fin_cases on all six indices. Each cell such as this one supplies the residual goal for one tuple.
In the gravity analysis stack this identity is bookkeeping for the 4D Regge midpoint M2TT kernel: once numerator and table agree up to the universal factor eight, later curvature and continuum-limit arguments can quote the closed form instead of the fold. It is pure discrete algebra on $\mathbb{F}_4$ indices, not a continuum GR claim by itself.
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